Regression modeling on fluvial networks focuses on capturing covariate effects on a target variable while considering the complex spatial heterogeneity of river systems. Traditional methods assume covariate effects are spatially constant, a limitation in understanding fluvial processes. To address this, we propose a spatially varying coefficient linear regression model that integrates concepts from Graph Signal Processing for nonparametric estimation. Our proposal introduces an estimator of a smooth covariance matrix-valued function over the network, allowing for local coefficient estimation. This framework is well-suited for environmental applications where local variability is critical, offering robust tools for analyzing spatial ecological data across fluvial systems. The proposed methodology is illustrated using examples from a river network in Piemonte, Italy.